Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Free products and the lack of state-preserving approximations of nuclear $ C^*$-algebras


Author: Caleb Eckhardt
Journal: Proc. Amer. Math. Soc. 141 (2013), 2719-2727
MSC (2010): Primary 46L05, 46L09
DOI: https://doi.org/10.1090/S0002-9939-2013-11702-8
Published electronically: April 8, 2013
MathSciNet review: 3056562
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ A$ be a homogeneous $ C^*$-algebra and $ \phi $ a state on $ A.$ We show that if $ \phi $ satisfies a certain faithfulness condition, then there is a net of finite-rank, unital completely positive, $ \phi $-preserving maps on $ A$ that tend to the identity pointwise. This, combined with results of Ricard and Xu, shows that the reduced free product of homogeneous $ C^*$-algebras with respect to these states has the completely contractive approximation property. We also give an example of a faithful state on $ M_2\otimes C[0,1]$ for which no such state-preserving approximation of the identity map exists, thus answering a question of Ricard and Xu.


References [Enhancements On Off] (What's this?)

  • 1. Marek Bożejko and Massimo A. Picardello.
    Weakly amenable groups and amalgamated products.
    Proc. Amer. Math. Soc., 117(4):1039-1046, 1993. MR 1119263 (93e:43005)
  • 2. Jean De Cannière and Uffe Haagerup.
    Multipliers of the Fourier algebras of some simple Lie groups and their discrete subgroups.
    Amer. J. Math., 107(2):455-500, 1985. MR 784292 (86m:43002)
  • 3. Kenneth J. Dykema.
    Faithfulness of free product states.
    J. Funct. Anal., 154(2):323-329, 1998. MR 1612705 (99e:46066)
  • 4. Kenneth J. Dykema.
    Exactness of reduced amalgamated free product $ C^*$-algebras.
    Forum Math., 16(2):161-180, 2004. MR 2039095 (2004m:46142)
  • 5. Eberhard Kirchberg.
    On subalgebras of the CAR-algebra.
    J. Funct. Anal., 129(1):35-63, 1995. MR 1322641 (95m:46094b)
  • 6. Éric Ricard and Quanhua Xu.
    Khintchine type inequalities for reduced free products and applications.
    J. Reine Angew. Math., 599:27-59, 2006. MR 2279097 (2009h:46110)
  • 7. Walter Rudin.
    Well-distributed measurable sets.
    Amer. Math. Monthly, 90(1):41-42, 1983. MR 691011 (84m:28001)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 46L05, 46L09

Retrieve articles in all journals with MSC (2010): 46L05, 46L09


Additional Information

Caleb Eckhardt
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47906
Email: caeckhar@math.purdue.edu

DOI: https://doi.org/10.1090/S0002-9939-2013-11702-8
Received by editor(s): November 15, 2010
Received by editor(s) in revised form: October 28, 2011
Published electronically: April 8, 2013
Additional Notes: A portion of this work was completed while the author was funded by the research program ANR-06-BLAN-0015 and by NSF grant DMS-1101144
Communicated by: Marius Junge
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society